Consider a filtered probability space \((\Omega , \mathcal {F}, (\mathcal {F}_t)_{t \in [0,T]}, \mathbb {P})\) , where \(\mathbb {P}\)  is the law of the super-Brownian motion on \(\mathbb {R}^d\) , \(d \ge 1\) , starting at \(m \in M_F(\mathbb {R}^d)\) , \(\mathcal {F}\)  is the Borel-\(\sigma \) -field corresponding to \(\Omega = C([0,T], M_F(\mathbb {R}^d))\)  and \((\mathcal {F}_t)_{t \in [0,T]}\)  is the canonical filtration satisfying the usual conditions. Also, denote the Laplacian operator by \(\Delta \) . From [1]}, we know that the probability measure \(\mathbb {P}\)  is a solution to the following martingale problem:
\(\mathbb {P}(X(0) = m) = 1 \text{ and for all $\phi \in D(\textstyle {\frac{1}{2}}\Delta )$ the process} \\M(t)(\phi ) = \langle X(t), \phi \rangle - \langle X(0) , \phi \rangle - \int _0^t \langle X(s), \frac{1}{2}\Delta \phi \rangle ds , \quad t \in [0,T] \\\text{is a $(\mathcal {F}_t)_t$-martingale $\mathbb {P}$-a.s. and has quadratic variation}\\[M(\phi )]_t = \int _0^t \langle X(s), \phi ^2 \rangle ds \quad \forall t \in [0,T] \quad \mathbb {P}-a.s..\) 
